For bounded C^{1,1} domains, every BV function u satisfies the trace inequality with constant 1 on the total variation term, and this fails for some C^{1,α} domains, 0<α<1.
Della Pietra, F
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Trace inequality for $BV(\Omega)$ in smooth domains
For bounded C^{1,1} domains, every BV function u satisfies the trace inequality with constant 1 on the total variation term, and this fails for some C^{1,α} domains, 0<α<1.